1. Field of the Invention
The present invention concerns a method for estimating the direction of arrival of a reference signal emitted by a signal source. In particular, the present invention can be used for estimating the direction of arrival of a signal emitted by a mobile terminal.
2. Discussion of the Background
Before reviewing the state of the art in the field of DOA (direction of arrival) estimation, the technique of passive beamforming will be shortly introduced.
An adaptive antenna generally comprises an antenna array and a beamformer as shown on FIG. 1. The antenna 100 may have an arbitrary geometry and the elementary sensors 1001, . . . , 100L may be of an arbitrary type. We consider an array of L omnidirectional elements immersed in the far field of a sinusoidal source S of frequency ƒ0. According to the far-field condition we may consider a plane wave arriving from the source in direction ("PHgr"0,xcex80). The first antenna is arbitrarily taken as the time origin. The travelling time difference between the l th element and the origin is given by:                                           τ            l                    ⁡                      (                                          ϕ                0                            ,                              θ                0                                      )                          =                                             less than                                                 r                  i                                →                                      ,                                                            u                  →                                ⁡                                  (                                                            ϕ                      0                                        ,                                          θ                      0                                                        )                                             greater than                                 c                                    (        1        )            
where {right arrow over (r)}l is the position vector of the l th element, {right arrow over (u)}("PHgr"0,xcex80) is the unit vector in the direction ("PHgr"0,xcex80), c is the speed of propagation, and  less than ,  greater than  represents the inner product. For a uniform linear array (ULA) i.e. a linear array of equispaced elements, with element spacing d aligned along the X-axis and the first element located at the origin (1) may be expressed as:                                           τ            l                    ⁡                      (                          θ              0                        )                          =                              d            c                    ⁢                      (                          l              -              1                        )                    ⁢          cos          ⁢                      xe2x80x83                    ⁢                      θ            0                                              (        2        )            
The signal received from the first element can be expressed in complex notation as:
m(t)exp(j2xcfx80f0t)xe2x80x83xe2x80x83(3)
where m(t) denotes the complex modulating function.
Assuming that the wavefront on the l th element arrives xcfx84l("PHgr"0,xcex80) later on the first element, the signal received by the l th element can be expressed as:
m(t)exp(j2xcfx80ƒ0(t+xcfx84l("PHgr"0,xcex80)))xe2x80x83xe2x80x83(4)
The expression is based upon the narrow-band assumption for array signal processing, which assumes that the bandwidth of the signal is narrow enough and the array dimensions are small enough for the modulating function to stay almost constant over xcfx84l("PHgr"0,xcex80), i.e. the approximation m(t)≅m(t+xcfx84l("PHgr"0,xcex80)) is valid. Then the signal received at the l th element is given by:
xl(t)=m(t)exp(j2xcfx80ƒ0(t+xcfx84l("PHgr"0,xcex80)))+nl(t)xe2x80x83xe2x80x83(5)
where nl(t) is a random noise component, which includes background noise and electronic noise generated in the lth channel. The resulting noise is assumed temporally white Gaussian with zero mean and variance equal to "sgr"2.
Passive beamforming consists in weighting the signals received by the different elements with complex coefficients xcfx89l and summing the weighted signals to form an array output signal. By choosing the complex coeffcients, it is possible to create a receiving pattern exhibiting a maximum gain in the direction xcex80 of the source. The array output signal can be expressed as the product of m(t) and what is commonly referred to as the array factor F:                               F          ⁡                      (            θ            )                          =                              ∑                          l              =              1                        L                    ⁢                                    ω              l                        ⁢                          exp              ⁡                              (                                                      j                    ⁡                                          (                                              l                        -                        1                                            )                                                        ⁢                  κ                  ⁢                                      xe2x80x83                                    ⁢                  d                  ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  θ                                )                                                                        (        6        )            
where xcexa=2xcfx80/xcex is the magnitude of the so-called wave-vector and xcex is the wavelength of the emitted signal. If we denote the complex weight xcfx89l=xcfx81lexp(jxcfx86l), the array factor can be written:                               F          ⁡                      (            θ            )                          =                              ∑                          l              =              1                        L                    ⁢                                    ρ              l                        ⁢                          exp              (                              j                ⁡                                  (                                                            φ                      l                                        +                                                                  (                                                  l                          -                          1                                                )                                            ⁢                      κ                      ⁢                                              xe2x80x83                                            ⁢                      d                      ⁢                                              xe2x80x83                                            ⁢                      cos                      ⁢                                              xe2x80x83                                            ⁢                      θ                                                        )                                                                                        (        7        )            
If we choose xcfx86l=xe2x88x92(lxe2x88x921)xcexad cos xcex80, the maximum response of F(xcex8) will be obtained for angle xcex80, i.e. when the beam is steered towards the wave source.
We suppose now that the array factor is normalized and that xcfx81l=1/L. If we consider an arbitrary direction of arrival xcex8.                               F          ⁡                      (            θ            )                          =                              1            L                    ⁢                                    ∑                              l                =                1                            L                        ⁢                          exp              ⁡                              (                                  j                  ⁢                                      xe2x80x83                                    ⁢                  κ                  ⁢                                      xe2x80x83                                    ⁢                                      d                    ⁡                                          (                                              l                        -                        1                                            )                                                        ⁢                                      (                                                                  cos                        ⁢                                                  xe2x80x83                                                ⁢                        θ                                            -                                              cos                        ⁢                                                  xe2x80x83                                                ⁢                                                  θ                          0                                                                                      )                                                  )                                                                        (        8        )            
that is, by denoting "psgr"=xcexad(cos xcex8xe2x88x92cos xcex80):                               F          ⁡                      (            ψ            )                          =                                            xe2x80x83                                      xe2x80x83                                ⁢                                    1              ⁢                              xe2x80x83                            ⁢                              sin                ⁡                                  (                                      L                    ⁢                                          xe2x80x83                                        ⁢                                          ψ                      /                      2                                                        )                                                                                    L                ⁢                sin                            (                              xe2x80x83                            ⁢                              ψ                /                2                            )                                ⁢                      exp            ⁡                          (                                                j                  ⁡                                      (                                          L                      -                      1                                        )                                                  ⁢                                  ψ                  /                  2                                            )                                                          (        9        )            
If we denote w=(xcfx891, . . . , xcfx89L)T the vector of the weighting coefficients used in the beamforming, the output of the beamformer can be simply expressed as wHx. The vector w for steering the beam in the look direction xcex80 is w=a(xcex80) where a(xcex80)=[1,exp(j xcexad cos xcex80), . . . , exp(j(Lxe2x88x921)xcexad cos xcex80)]T. If RN="sgr"2I is the covariance matrix of the uncorrelated noise, the power of the noise component at the array output may be written:                               P          N                =                                            w              H                        ⁢                          R              N                        ⁢            w                    =                                    σ              2                        L                                              (        10        )            
In other words, the noise power at the array output is 2/Lth the noise power present at each element. Thus, beamforming with unity gain in the signal direction has reduced the uncorrelated noise by a factor L and thereby increased the output signal to noise ratio (SNR).
Turning now to a more general case where M point sources are present in the far field, the signal received by an element l can be written:                                           x            l                    ⁡                      (            t            )                          =                                            ∑                              m                =                1                            M                        ⁢                                                            s                  m                                ⁡                                  (                  t                  )                                            ⁢                              exp                ⁡                                  (                                                            j                      ⁡                                              (                                                  l                          -                          1                                                )                                                              ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                                          θ                      m                                                        )                                                              +                                    n              l                        ⁡                          (              t              )                                                          (        11        )            
If we consider the sampled signals at sampling times nT, n={1, . . . , N}, denoted n for sake of simplification (11) can be rewritten:                               x          ⁡                      (            n            )                          =                                            ∑                              m                =                1                            M                        ⁢                                          a                ⁡                                  (                                      θ                    m                                    )                                            ⁢                                                s                  m                                ⁡                                  (                  n                  )                                                              +                      n            ⁡                          (              n              )                                                          (        12        )            
where a(xcex8m)=[1,exp(jxcfx80 cos xcex8m), . . . , exp(j(Lxe2x88x921)xcfx80 cos xcex8m)]T is a vector called the array response, x(n) is the vector of the received signals at time n and n(n)=[n1(n),n2(n), . . . , nL(n)]T is the noise vector. The sampled array output x(n) can be expressed as a matrix product:
x(n)=As(n)+n(n)xe2x80x83xe2x80x83(13)
where A=[a(xcex81),a(xcex82), . . . , a(xcex8M)] is the Lxc3x97M matrix the columns of which are the vectors a(xcex8m) and s(n)=[s1(n),s2(n), . . . , sM(n)]T is the signal vector.
We suppose that the signals and noise samples are stationary and ergodic complex-valued random processes with zero mean, uncorrelated with the signals and uncorrelated each other. They are modeled by temporally white Gaussian processes and have identical variance "sgr"2.
Most of the DOA estimation techniques are based upon the calculation of an estimate of the spatial covariance matrix R:                     R        =                              E            ⁡                          [                                                x                  ⁡                                      (                    n                    )                                                  ⁢                                                      x                    H                                    ⁡                                      (                    n                    )                                                              ]                                =                                    lim                              N                →                ∞                                      ⁢                                          1                N                            ⁢                                                ∑                                      n                    =                    1                                    N                                ⁢                                                      x                    ⁡                                          (                      n                      )                                                        ⁢                                                            x                      H                                        ⁡                                          (                      n                      )                                                                                                                              (        14        )            
which can be rewritten according to the matrix notation of (13):
R=APAH+"sgr"2Ixe2x80x83xe2x80x83(15)
where P is the source covariance matrix and I is the identity matrix.
Not surprisingly, since R reflects the (spatial) spectrum of the received signal and the directions of arrival are obviously linked with the peaks of the spectrum, most of the DOA estimation techniques make use of the spectral information contained in R.
The simplest; DOA estimation technique (also called conventional DOA estimation) merely amounts to finding the peaks of the spatial spectrum:                               P                      B            ⁢                          xe2x80x83                        ⁢            F                          =                                                            a                H                            ⁡                              (                θ                )                                      ⁢            R            ⁢                          xe2x80x83                        ⁢                          a              ⁡                              (                θ                )                                                          L            2                                              (        16        )            
i.e. the maximum output power when the beam is steered over the angular range. However, this method suffers from severe resolution limitation when a plurality of sources are present.
Numerous DOA estimation techniques have been designed in the prior art and no purpose would be served here by reciting them all. A review of these techniques can be found in the article of H. Krim and M. Viberg entitled xe2x80x9cTwo decades of array signal processing researchxe2x80x9d published in IEEE Signal Processing Magazine, July 1996, pp. 67-74.
The most popular DOA estimation techniques are MUSIC (MUltiple Signal Classification) and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) as well as various techniques derived therefrom.
The MUSIC algorithm relies on an eigenanalysis of the spatial covariance matrix R. The array reponse vectors a(xcex8) define an Hilbertian space of dimension L which can be decomposed into a signal sub-space of dimension M and a noise sub-space of dimension Lxe2x88x92M (it is assumed that M less than L). R is a hermitian, positive semi-definite matrix of rank M (the M sources are supposed uncorrelated and therefore P is full rank). The eigenvalues of R can be ranked: xcex1xe2x89xa7xcex2xe2x89xa7 . . .  greater than xcexM+1=. . . =xcexL =xcfx842 where the first M eigenvalues correspond to eigenvectors spanning the signal sub-space and the last Lxe2x88x92M eigenvalues correspond to eigenvectors spanning the noise subspace. Denoting:
IIxe2x8axa5=Ixe2x88x92A(AHA)xe2x88x921AHxe2x80x83xe2x80x83(17)
the projection operator onto the noise sub-space, the directions of arrival are determined by finding the peaks in the so-called MUSIC spectrum:                                           P            M                    ⁡                      (            θ            )                          =                                                            a                H                            ⁡                              (                θ                )                                      ⁢                          xe2x80x83                        ⁢                          a              ⁡                              (                θ                )                                                                                        a                H                            ⁡                              (                θ                )                                      ⁢                          Π              ⊥                        ⁢                          xe2x80x83                        ⁢                          a              ⁡                              (                θ                )                                                                        (        18        )            
Various improvements of the MUSIC algorithm have been proposed in the litterature in order to overcome some shortcomings in specific measurement conditions.
The ESPRIT algorithm exploits the rotational invariance of the signal sub-space when the antenna array is invariant by translation (e.g. a ULA). A detailed description of the algorithm can be found in the article of R. Roy et al. entitled xe2x80x9cESPRIT: Estimation of Signal Parameters via Rotational Invariance Techniquesxe2x80x9d published in IEEE Transactions on ASSP, Vol. 37, No. 7, July 1989, pp. 984-995. Here again, the algorithm relies on an eigendecomposition of the array covariance matrix R leading to a decomposition into a signal sub-space and a noise subspace.
Furthermore, the DOA estimation methods based on signal and noise sub-space decomposition, e.g. MUSIC or ESPRIT, requires the knowledge of the number M of signal sources. This number may be obtained from the multiplicity of the eigenvalue "sgr"2 in the covariance matrix. In most cases, however, the value "sgr"2 is unknown and M is derived from the number of xe2x80x9cmost equalxe2x80x9d eigenvalues which is also called the MDL (Minimum Description Length) criterion.
The above mentioned algorithms use the covariance matrix R. In practice, of course, this matrix is not avaible and must be estimated e.g. by expression (14). The noise level on the coefficients of the estimate of the covariance matrix may lead to an erroneous determination of the DOAs and/or number of signal sources.
It is an object of the invention to reduce the noise level on the estimate of the covariance matrix so as to improve the DOA and/or number of sources estimation especially for low signal-to-noise ratios (SNRs).
This problem is solved by the method defined in appended claim 1 i.e. by a method for estimating a direction of arrival of a signal transmitted by a signal source and received by an antenna array, said method estimating the covariance matrix of the signals respectively received by the antennas of the array and deriving therefrom a direction of arrival, wherein said received signals are correlated with a reference signal transmitted by said source prior to be submitted to the covariance matrix estimation.